Distribution of tree parameters by martingale approach

TL;DR

This paper investigates the limiting distributions of various parameters in uniform random labeled trees, using martingale methods and the Aldous-Broder algorithm, revealing normal and log-normal asymptotic behaviors.

Contribution

It introduces a martingale-based framework to analyze the asymptotic distribution of tree parameters, including pattern occurrences and automorphisms, in random labeled trees.

Findings

Number of small pattern occurrences is asymptotically normal.

Number of automorphisms is asymptotically log-normal.

Provides a unified approach for analyzing local perturbation stability.

Abstract

For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem and the Aldous--Broder algorithm. In particular, our general result implies the asymptotic normality of the number of occurrences of any given small pattern and the asymptotic log-normality of the number of automorphisms.